Use of capacitors to regulate the voltage in the network
Seminar paper
Power Distribution and Industrial Systems
Mentor:
Prof. dr. Grega Bizjak, univ. dipl. inž. el.
Student:
Gentrit Rexha
Ljubljana, April, 2018
Table of Contents
Table of Figures ............................................................................................................................................. 3
1
Power System ....................................................................................................................................... 4
2
Production and Consumers of Reactive Power .................................................................................... 5
2.1
2.1.1
Synchronous Machines ......................................................................................................... 6
2.1.2
Static compensators – Capacitors ......................................................................................... 7
2.2
Consumers of Reactive Power ...................................................................................................... 8
2.2.1
Asynchronous Motor ............................................................................................................ 9
2.2.2
Transformer .......................................................................................................................... 9
2.3
3
Production of Reactive Power ...................................................................................................... 6
Power Lines ................................................................................................................................. 10
Voltage drop........................................................................................................................................ 11
3.1
Calculation of Voltage Drop on a Line......................................................................................... 12
3.1.1
Ferranti effect ..................................................................................................................... 14
3.1.2
Tab Transformers for voltage regulation ............................................................................ 15
3.2
Power Factor correction ............................................................................................................. 16
3.2.1
4
Concept of Leading and Lagging Power Factors ................................................................. 17
Capacitors in Power Systems .............................................................................................................. 18
4.1
Line Compensation ..................................................................................................................... 19
4.2
Serial Capacitors.......................................................................................................................... 20
4.2.1
4.3
Shunt Capacitors ......................................................................................................................... 25
4.3.1
4.4
Overcompensation.............................................................................................................. 23
Capacitor installation types ................................................................................................ 27
Power factor correction .............................................................................................................. 30
5
Questions ............................................................................................................................................ 32
6
Homework task ................................................................................................................................... 32
7
Conclusion ........................................................................................................................................... 35
8
References .......................................................................................................................................... 35
2
Table of Figures
Figure 1.1 System frequency dependence from the supply and demand ......................................................................4
Figure 2.1 Capability curve of a generator ....................................................................................................................7
Figure 2.2 Reactive power characteristic from capacitive banks on the voltage level connected .................................8
Figure 2.3 Daily load diagram of active and reactive power .........................................................................................9
Figure 3.1 Equivalent scheme of a distribution line .....................................................................................................12
Figure 3.2 The phasor diagram representation of voltage drop in a line ....................................................................12
Figure 3.3 Phase diagram of voltage drop of a loaded lined .......................................................................................13
Figure 3.4 Ferranti effect on a power line ....................................................................................................................14
Figure 3.5 Representation scheme for lines shorter than 80 km .................................................................................15
Figure 3.6 Representation scheme for lines longer than 80 km...................................................................................15
Figure 3.7 Examples of some of the sources of leading and lagging reactive power at the load ................................17
Figure 3.8 Reactive power meter .................................................................................................................................18
Figure 4.1 Capacitor bank ............................................................................................................................................19
Figure 4.2 Voltage phasor diagrams for a line with a lagging power factor load (a) and (c) and leading power factor
(b) and (d) ....................................................................................................................................................................21
Figure 4.3 Mechanical switching arrangement to adjust the capacitance of the capacitive banks ............................23
Figure 4.4 Overcompensation of the end voltage: (a) at normal load and (b) at the start of a large motor ..............24
Figure 4.5 Voltage phasor diagram with leading power factor: (a) without series capacitors and (b) with series
capacitors ....................................................................................................................................................................24
Figure 4.6 Voltage profile along a feeder with capacitors ...........................................................................................25
Figure 4.7 Voltage phasor diagrams for a feeder circuit of a lagging power factor: (a) and (c) without compensation
and (b) and (c) with shunt capacitor compensation. ...................................................................................................26
Figure 4.8 The effects of the fixed capacitor on the voltage profile of (a) feeder with uniformity load, (b) at heavy
load and (c) at light load..............................................................................................................................................28
Figure 4.9 Sizing of the fixed and switched capacitors to meet the daily reactive power demand .............................30
Figure 4.10 (a) Phasor diagram and (b) Power triable for a typical distribution load .................................................30
Figure 4.11 Illustration of (a) the use of a power triangle for the power factor correction by employing capacitive
reactive power and (b) the required increase in the apparent and reactive powers as a function of the load power
factor, holding the real power of the load constant. ...................................................................................................31
Figure 4.12 Illustration of power factor correction by employing a shunt capacitor ..................................................32
Figure 6.1 Power triangle ............................................................................................................................................34
3
1 Power System
The function of power systems consists the generation, transmission, distribution,
regulation and consumption of the electrical energy. Since the accumulation of energy is
not a good economical solution, the process of transmission, distribution and
consumption it is realized in the same moment. Power systems that lie in a region, state
or continent are interconnected with each other and operate as a single unit which is
maximally harmonized and synchronized. The production and consumption of active and
reactive power has to be in balance.
Balance of the production and consumption restrict the self-regulation of the power
systems. If the load of the power system would quite increase and if this increase could
not be supplied by the power system than this would cause the degradation of the quality
of electrical energy, respectively the decrease of the systems frequency and voltage. This
would cause the consumers to withdraw less energy from the power system with what
the demand would decrease and the power system would stabilize in a lower frequency.
However, there is only a few percent margin for such self-regulation.
If voltage is propped up with the injection of reactive power, then the demand increases
more than supplied by transmitting lines will cause a consequent drop of frequency that
may result in system collapse. Alternatively, if there is inadequate reactive power, the
system may have voltage collapse. Voltage collapses usually occur on power system
which are heavily loaded or faulted or have shortage of reactive power. Voltage collapse
is a system instability involving many power system components. In fact, a voltage
collapse may involve an entire power system. Voltage collapse is typically associated with
reactive power demand of load not being met due to shortage in reactive power
production and transmission. Voltage collapse is a manifestation of voltage instability in
the system.
System operators use reactive power resources to maintain the voltage at all the buses
around the nominal value. Keeping transmission level voltages at nominal value or within
a tight range ensures proper voltages at the distribution levels. Another important factor
is that the transmission network security is closely associated with the voltage profile.
Since the voltage on a bus is strongly coupled with the supply of reactive power, the
voltage control service is also called reactive power support service. It is prudent to control
the bus voltages by providing reactive power locally, rather than making it to flow through
the grid. There are three major reasons
for this. First, the power system
equipment is designed to operate within
a range of voltages, usually within ± 5%
of the nominal voltage. At low voltages,
the performance of most of the electrical
equipment’s is poor. For example, Figure 1.1 System frequency dependence from the supply and
demand
4
induction motors can overheat and get damaged. High voltages will not only damage the
equipment but also will short their life.
Second, the power transmission capability available from a transmission line design is
limited by technological as well as economical constraints. The reactive power consumes
transmission and generation capacity. To maximize the amount of real power that can be
transferred across a congested transmission interface, reactive power flows must be
minimized. Similarly, reactive power production can limit a generator’s real power
capability. Third, moving reactive power on the transmission system incurs real power
losses. Thus, additional energy must be supplied to replace these losses.
2 Production and Consumers of Reactive Power
The reactive power flow on the system is reflected negatively on the voltage profile on the
power system. Even though these values are tried to be reduced, their presence on the
system is unavoidable, this because all the elements on the system need reactive power
for their own needs. These needs are especially important for some consumers which
their demand change significantly during the time.
Some system operators pay generators their embedded costs for reactive resources.
However, determining the embedded costs of generator to provide reactive power support
leads to ambiguity. This is so because; the same equipment is used to provide both real
and reactive power. Questions like what percentages, for example, of the exciter,
generator stator, generator rotor, turbine assembly, and step-up transformer should be
assigned to each function are not easy to answer. Table 2.1 shows the comparison of
various types of reactive power sources.
Table 2.1 Comparison of various types of reactive power sources
5
2.1 Production of Reactive Power
The main producers of reactive power in the power system are the synchronous
generators, synchronous compensators and static compensators in the form of capacitive
batteries. Beside of these, also the lines in certain circumstances appear as producers of
the reactive power. It is important to mention that for the synchronous generator, the
production of reactive power restricts the production of active power which also decreases
the economical effectivity of the power plant.
2.1.1
Synchronous Machines
Synchronous machines that are designed exclusively to provide reactive support are
called synchronous condensers. Synchronous Condenser in reality is just a synchronous
machine which works as a synchronous motor without a mechanical load or a
synchronous generator. Their power factor is not exactly zero, because these machines
withdraw from the power grid the needed power to cover their no load losses. When such
a machine works as overexcited it produces reactive power and when it works under
excited it appears on the grid as a costumer of the reactive load. This is why synchronous
compensators are appropriate for voltage regulation if we look from the technical aspect,
but their application is limited from the cost.
The synchronous generator also can work as a synchronous compensator. During the
low loads that usually happen at night time the synchronous generator operates as an
under excited machine and appears on the grid as a consumer of the reactive power that
is generated from the high voltage lines and this way it contributes by decreasing the
voltages in the system which sometimes can exceed their allowed values.
The synchronous generators are very fast reactive support devices. The ability of a
generator to provide reactive support depends on its real-power production. Figure 2.1
shows the limits on real and reactive production for a typical generator. This is also called
as a capability curve of a generator. Like most electric equipment, generators are limited
by their current-carrying capability. Near rated voltage, this capability becomes an MVA
limit for the armature of the generator rather than a MW limitation, shown as the armature
heating limit in the figure. Production of reactive power involves increasing the magnetic
field to raise the generator’s terminal voltage. Increasing the magnetic field requires
increasing the excitation current in the rotating field winding. This too is current limited,
resulting in the field-heating limit shown in the figure. Absorption of reactive power is
limited by the magnetic-flux pattern in the stator, which results in excessive heating of the
stator-end iron, the core-end heating limit. The synchronizing torque is also reduced when
absorbing large amounts of reactive power, which can also limit generator capability to
reduce the chance of losing synchronism with the system.
6
During the nominal mode, the regulation consists on the increase of the reactive power
of the generator, or his voltage when it comes up to the increase of the active load, while
with the decrease of his reactive power, or voltage when his active power is decreased.
Figure 2.1 Capability curve of a generator
2.1.2
Static compensators – Capacitors
Capacitors and inductors are passive devices that generate or absorb reactive power.
They accomplish this without significant real-power losses or operating costs. The output
of capacitors and inductors is proportional to the square of the voltage. Capacitor banks
are composed of individual capacitors. The individual capacitors are connected in series
and parallel to obtain the desired capacitor-bank voltage and capacity rating. The
capacitor banks are often configured with several steps to provide a limited amount of
variable control.
Capacitors are used as capacitive batteries and produce reactive power. Their U-Q
characteristic is pressed in the figure 2.2. From the following equation shows the reactive
power produced from a capacitor, we can see that with the decrease of the voltage we
will have a lower rate of reactive power produced and vice versa.
𝑄𝐶 = 3𝜔𝐶𝑈 2
This is consider as one of the technical disadvantages, beside of that their application in
the distribution networks is quite convenient and quite spread. Actually capacitive
batteries represent the main device for voltage and reactive power regulation in the
transmission and distribution power system.
7
Reactive Power
Q-U characteristic
Voltage
Figure 2.2 Reactive power characteristic from capacitive banks on the voltage level connected
Capacitors provide tremendous benefits to distribution system performance. Most noticeably,
capacitors reduce losses, free up capacity, and reduce voltage drop:


Losses; Capacity - By canceling the reactive power to motors and other loads with low
power factor, capacitors decrease the line current. Reduced current frees up capacity; the
same circuit can serve more load. Reduced current also significantly lowers the I 2R line
losses.
Voltage drop - Capacitors provide a voltage boost, which cancels part of the drop caused
by system reactive loads. Switched capacitors can regulate voltage on a circuit.
If applied properly and controlled, capacitors can significantly improve the performance of
distribution circuits. But if not properly applied or controlled, the reactive power from capacitor
banks can create losses and high voltages. The greatest danger of overvoltages occurs under
light load. Good planning helps ensure that capacitors are sited properly. More sophisticated
controllers reduce the risk of improperly controlling capacitors like SVC – Static VAr
Compensation and STATCOM - Static Synchronous compensator.
2.2 Consumers of Reactive Power
The required reactive power can be represented by the daily reactive load diagram as it
is represented in figure 2.3. If we compare the daily diagram of active and reactive load
we can see that for the active load consumption during 24 hours there are two peaks, one
in the morning and one in the evening, in the other hand for the reactive load consumption
we can see that there is only one peak which one decreases after the end of the first shift
of the industrial consumers. These consumers have the most needs for relative power.
For this reason the flow of reactive power through the system elements that are mainly
reactive consumers will cause high voltage drops on the system.
8
Figure 2.3 Daily load diagram of active and reactive power
2.2.1
Asynchronous Motor
One of the main consumer of reactive power are the asynchronous motors. These motors
even in the no load operation they withdraw reactive power from the system round 30%
of their nominal power. With the increase of load, the active power consumption is
increased and with that the consumption of reactive power which can go up to 50%. The
equation that represents the reactive power consumption is described below:
2
𝑄𝐴𝑀
𝑃 2
𝑃𝑛
= 𝑄0 + 𝑝 ∆𝑄𝑛 = 3𝑈𝑛 𝐼0 + ( ) 3 (
) 𝑋𝑚
𝑃𝑛
√3 𝑈𝑛
2
When we consider that the area of application of the asynchronous motor in the industry
where usually it takes more than 60% of the total consumption we can imagine what are
the values of the reactive power required.
2.2.2
Transformer
Transformers are largest consumes of the reactive power in the group of powers systems
elements. In percentage to the nominal power the consumption of the transformer is
significantly smaller when compared with the asynchronous motor. Unlike the
asynchronous machine, the transformer in no load operation it consumes only 1- 2%
reactive power of its nominal power. A considerable value of reactive power is spent by
the current of the loads that flow through the transformers reactance. The following
equation shows the consumption of reactive power of a transformers during different
loads:
2
𝑄𝑡 = 𝑄0 + 𝑠 𝑄𝑛 = 3𝑈𝑛 𝐼0 + 3
𝐼𝑛2
𝑆 2
( )
𝑆𝑛
9
For nominal load of the transformer the reactive power consumption goes up to 10% of
its nominal power. Even though this value is relatively low we have to consider that the
energy has to be transferred from the production line to the consumer, what means that
it has to pass 4-5 transformations and in each transformation will have its own
consumption or losses. Practiclly, to deliver 1 MW of active power, we have to spend 0.4
up to 0.5 MVAr of reactive power. So we can conclude that transformers are a
considerable reactive consumers in the power system.
2.3
Power Lines
Electrical power lines, mounted underground or in air, can appear as consumers of
reactive power when they are loaded more than their natural power. But they can appear
as a producer of reactive power in the case when they are loaded under their natural
power, this phenomena is known as Ferranti effect. In these cases the reactive power
that is spent in the longitudinal reactance of the line is generated or compensated from
the transversal susceptance of the line. Transmission lines produce reactive power
(MVAr) due to their natural capacitance. The amount of MVAr produced is dependent on
the transmission lines capacitive reactance XC and the voltage at which line is energized.
In equation form the MVAr produced by a transmission line is:
∆𝑄𝐶 =
𝑈2
𝑋𝐶
Transmission lines also use reactive power to support their magnetic fields. Magnetic
field strength is dependent on the magnitude of the current flow through the line and the
line’s natural inductive reactance XL. It follows then that the amount of MVAr used by a
transmission line is a function of current flow and inductive reactance. In equation form
the MVAr used by a transmission line is:
∆𝑄𝐿 = 3 𝐼 2 𝑋𝐿
The balance of reactive power in a power line is known as Surge Impedance Loading
(SIL) of a transmission line. When SIL occurs QC=QL than the surge impedance is
equal to:
𝑈2
3 𝐼 𝑋𝐿 =
𝑋𝐶
3 𝑈𝑓2
𝑋𝐿 𝑋𝐶 =
3 𝐼2
2
𝑈𝑓
𝜔𝐿
√
=
= 𝑍0
𝜔𝐶
𝐼𝑓
10
The above equation is known as the “surge impedance”. The significance of the surge
impedance is that if a purely resistive load that is equal to surge impedance is connected
to the end of a transmission line with no resistance, a voltage surge introduced to the
sending point of the line would be absorbed completely at the receiving point. The voltage
at the receiving point would have the same magnitude as the sending point voltage and
would have a phase angle that is lagging with respect to the sending point by an amount
equal to the time required to travel across the line from sending to receiving point. The
value of SIL to a System Operator is realizing that when a line is loaded above its SIL it
acts like a reactor – absorbing MVAr from the system and when a line is loaded below its
SIL it acts like a capacitor – supplying MVAr to the system.
When the following equation is equal to 0 we say that the reactive load on the line is
balanced otherwise the reactive power that is consumed or produced on a line is
represented with the following equation:
∆𝑄 = ∆𝑄𝐿 − ∆𝑄𝐶 = 3𝐼 2 𝑋𝐿 −
𝑈2
𝑋𝐶
If consider that the voltage 𝑈 ≈ 𝑐𝑜𝑛𝑠𝑡., from the abovementioned equation we can
conclude that for higher loads the reactive power that is spent in the longitudinal reactance
is higher than the reactive power generated from the transversal susceptance, therefore
the line appears like a reactive power consumer:
∆𝑄𝐿 > ∆𝑄𝐶
For loads smaller than the natural power of the lines, this randomly happens during the
night time, especially when the lines are not loaded than the lines appear as generators
of the reactive power on the power system.
∆𝑄𝐿 < ∆𝑄𝐶
Practiclly, Surge Impedance of cables can be quite high (higher than the thermal limits)
and they are typically loaded much below their Surge Impedance Loading. This leads to
significant generation of reactive power and overvoltage problems if cable length is
greater than 80 km.
3 Voltage drop
The regulation of voltage and reactive power flow in the power system it is very important.
By level of importance it comes just after the regulation of frequency and active power.
Even though this two regulation types are connected with each other in some portion,
however they are discussed separately. This because the dependence between voltage
and reactive power is exposed as the dependence of frequency with active power.
11
Despite the frequency that in every point of the power system is the same, the voltage
practically in every point of the system is different, that’s because for every power flow or
current flow in the system will cause an unavoidable voltage drop.
3.1 Calculation of Voltage Drop on a Line
Let us consider the one radial line in which flows the power S=P+jQ in one way as it is
schematically presented in the figure 3.1. This line is loaded with a three phase
symmetrical load. If the voltage in the beginning of the line is written as U1 and the voltage
on the end U2 than the difference of the voltage from the beginning of the line with the
one in the end gives us the voltage drop on the line.
Figure 3.1 Equivalent scheme of a distribution line
In the figure 3.2 the voltage drop is expressed with the vector AB. Where the AC
represents the longitudinal vector and CB represents the transversal vector.
Figure 3.2 The phasor diagram representation of voltage drop in a line
12
From the phasor diagram in figure 3.2 we can see that this relation can be written as:
2
2
𝑈12 = (𝑈2 + 𝑈𝑙𝑜𝑛𝑔 ) + ∆𝑈𝑡𝑟𝑎𝑛𝑠
= (𝑈2 + 𝑅𝐼 𝑐𝑜𝑠𝜑 + 𝑋𝐼 𝑠𝑖𝑛𝜑)2 + (𝑋𝐼 𝑐𝑜𝑠𝜑 − 𝑅𝐼 𝑠𝑖𝑛𝜑)2
By knowing the relation for power we can write the voltage drop equation by the power
flow:
𝐼 𝑐𝑜𝑠𝜑 =
𝑃
√3𝑈
and
𝐼 𝑠𝑖𝑛𝜑 =
𝑄
√3𝑈
After the replacement on the we will have:
𝑈12 = (𝑈2 +
𝑃𝑅 + 𝑄𝑋 2
𝑃𝑋 − 𝑄𝑅 2
) +(
)
𝑈2
𝑈2
When we write in the vectorial form we have:
𝑈1 = (𝑈2 +
𝑃𝑅 + 𝑄𝑋
𝑃𝑋 − 𝑄𝑅
) + 𝑗(
)
𝑈2
𝑈2
.
Utrans
.
Ulong
.
Figure 3.3 Phase diagram of voltage drop of a loaded lined
The transversal component can be neglected because its value is insignificant against
the longitudinal component since the resistance is quite small against the reactance, so
the equation will take this form:
𝑈1 = (𝑈2 +
𝑃𝑅 + 𝑄𝑋
)
𝑈2
13
The transsmision lines are almost all reactive, so 𝑋 ≫ 𝑅. For the voltage drop we can
write:
∆𝑈 = 𝑈1 − 𝑈2 =
𝑄𝑋
𝑈2
And by this we can conclude that the voltage drop on the line it is dependent from the
flow of the reactive power through the line considering that the flow of active power is
unavoidable. By this equation we see that it exists a strong relation between the voltage
drop and the reactive power.
∆𝑈 ~ 𝑄
(8)
3.1.1 Ferranti effect
The effect in which the voltage at the end of the transmission line is higher than the
voltage in the beginning of the line is known as the Ferranti effect. Such type of effect
mainly occurs because of light load or open circuit at the receiving end.
Figure 3.4 Ferranti effect on a power line
Capacitance and inductance are the main parameters of the lines having a length 80 km
or above. On such transmission lines, the capacitance is not concentrated at some
definite points. It is distributed uniformly along the whole length of the line. At power
system frequencies, many useful simplifications can be made for lines of typical lengths.
For analysis of power systems, the distributed resistance, series inductance, shunt
leakage resistance and shunt capacitance can be replaced with suitable simplified
networks.
14
A short length of a power line (less than 80 km) can be approximated with a resistance in
series with an inductance and ignoring the shunt admittances. This value is not the total
impedance of the line, but rather the series impedance per unit length of line. For a longer
length of line (80–250 km), a shunt capacitance is added to the model. In this case it is
common to distribute half of the total capacitance to each side of the line ( model) or in
the middle of the line and distribute the resistance and reactance (T model). As a result,
the power line can be represented as a two-port network, such as ABCD parameters.
Figure 3.5 Representation scheme for lines shorter than 80 km
Figure 3.6 Representation scheme for lines longer than 80 km
The Ferranti Effect will be more pronounced the longer the line and the higher the voltage
applied. The relative voltage rise is proportional to the square of the line length.
The Ferranti effect is much more pronounced in underground cables, even in short
lengths, because of their high capacitance. It was first observed during the installation of
underground cables in Sebastian Ziani de Ferranti's 10 kV distribution system in 1887.
Now days the Ferranti effect is controlled by installing switchable reactance banks.
3.1.2 Tab Transformers for voltage regulation
Transformers are used to transfer power between different voltage levels or to regulate
real or reactive flow through a particular transmission corridor. Most transformers come
equipped with taps on the windings to adjust either the voltage transformation or the
reactive flow through the transformer. Such transformers are called either load-tapchanging (LTC) transformers or tap-changing-under-load (TCUL) transformers.
A tap changer is a mechanism in transformers which allows for variable turn ratios to be
selected in discrete steps. Transformers with this mechanism obtain this variable turn
ratio by connecting to a number of access points known as taps along the primary high
voltage and low current winding. These systems usually possess 1 tap per 1% of the
rated and allow a regulation in a rage from ±5% up to ±20% variation from the nominal
transformer rating which, in turn, allows for stepped voltage regulation of the output.
This method is applied for the regulation of the transformers on the power system. This
voltage regulation method is quite effective for the regulation of voltage on the bus bars
that don’t suffer much load changes. This method is based on changing the number of
windings of the transformers to minimize the current handling requirements of the
15
contacts. By this we can increase the voltage on the secondary side and by this we will
partly compensate the voltage drops on the system. This method is applied in the
middle and high voltage levels (35 kV and up) by installing the transformers in the peak
point of the system.
3.2
Power Factor correction
Most loads in modern electrical distribution systems are inductive. Examples include
motors, transformers, gaseous tube lighting ballasts, and induction furnaces as
represented in table 3.1.
Table 3.1 Power factor for common devices
Inductive or reactive loads, such as with capacitors or inductors, storage energy in the
loads that results in a phase difference between the current and voltage waveforms.
During each cycle of the AC voltage, extra energy, in addition to any energy consumed
in the load, is temporarily stored in the load in electric or magnetic fields, and then
returned to the power grid a fraction of the period later. So we can say that loads require
two kinds of currents or energies:
 Active or working power (kW) to perform the actual work of creating heat, light,
motion, machine output, and so on.
 Reactive power (kVAr) to sustain the magnetic field. Reactive power does not
perform useful “work,” but circulates between the generator and the load. It places
a heavier drain on the power source, as well as on the power source’s distribution
system.
16
The power factor of a power system is defined as the ratio of the real power flowing to the
load to the apparent power in the circuit, and is a dimensionless number in the closed
interval of −1 to 1. A high power factor signals efficient utilization of electrical power, while
a low power factor indicates poor utilization of electrical power. To determine power factor
(PF), divide working power (kW) by apparent power (kVA). In a linear or sinusoidal
system, the result is also referred to as the cosine.
However distribution companies typically charge industrial customers a penalty for power
factor lower than something like 0.85 or 0.9. For other customers, there is typically no
charge. Losses on the customer side of the meter are usually small enough to make it not
worthwhile to use sophisticated correction schemes.
3.2.1 Concept of Leading and Lagging Power Factors
Many consider that the terms “lagging” and “leading” power factor are somewhat
confusing, and they are meaningless, if the directions of the flows of real and reactive
powers are not known. In general, for a given load, the power factor is lagging (positive)
if the load withdraws reactive power; on the other hand, it is leading (negative) if the load
supplies reactive power. Hence, an induction motor has a lagging power factor since it
withdraws reactive power from the source to meet its magnetizing requirements. But a
capacitor (or an overexcited synchronous motor) supplies reactive power and thus has a
leading power factor, as shown in Figure 3.4:
Figure 3.7 Examples of some of the sources of leading and lagging reactive power at the load
17
On the other hand, an underexcited synchronous motor withdraws both the real and
reactive power from the source, as indicated. The use of varmeters instead of power
factor meters avoids the confusion about the terms “lagging” and “leading.” Such a
varmeter has a zero center point with scales on either side, one of them labeled “in” and
the other one “out.”
4 Capacitors in Power Systems
Figure 3.8 Reactive power meter
For the reduction of cost and improved reliability, most of the world’s electric power
systems continue to be interconnected. Interconnections take advantage of diversity of
loads, availability of sources and fuel price for supplying power to loads at minimal cost.
Compensation in power systems is, therefore, essential to alleviate some of these
problems. Series/shunt compensation has been in use for the past many years to achieve
this objective.
Load compensation is the management of reactive power to improve power quality i.e.
voltage profile and power factor. The reactive power flow is controlled by installing shunt
compensating devices (capacitors/reactors) at the load end bringing about proper
balanced between generated and consumed reactive power.
On power systems, capacitors do not store their energy very long—just one-half cycle.
Each half cycle, a capacitor charges up and then discharges its stored energy back into
the system. The net real power transfer is zero. Just when a motor with low power factor
needs power from the system, the capacitor is there to provide it. Then, in the next half
cycle, the motor releases its excess energy, and the capacitor is there to absorb it.
Capacitors and reactive loads exchange this reactive power back and forth. This benefits
the system because that reactive power (and extra current) does not have to be
transmitted from the generators all the way through many transformers and many
kilometers of lines; the capacitors can provide the reactive power locally. This frees up
the lines to carry real power, power that actually does work.
Capacitor units are made of series and parallel combinations of capacitor packs or
elements put together as shown in Figure 4.1.
18
Figure 4.1 Capacitor bank
Capacitors are made within a given tolerance. The IEEE standard allows reactive power
to range between 100% and 110% when applied at rated sinusoidal voltage and
frequency (at 25°C case and internal temperature) (IEEE Std. 18-2002). In practice, most
units are from +0.5% to +4.0%, and a given batch is normally very uniform. Capacitor
losses are typically on the order of 0.07 to 0.15 W/kVAr at nominal frequency.
Losses include resistive losses in the foil, dielectric losses, and losses in the internal
discharge resistor. Capacitors must have an internal resistor that discharges a capacitor
to 50 V or less within 5 min when the capacitor is charged to the peak of its rated voltage.
This resistor is the major component of losses within a capacitor.
Capacitors have very low losses, so they run very cool. But capacitors are very sensitive
to temperature and are rated for lower temperatures than other power system equipment
such as cables or transformers. Also, capacitors are designed to operate at high dielectric
stresses, so they have less margin for degraded insulation. Standards specify an upper
limit for application of 40°C or 46°C depending on arrangement. These limits assume
unrestricted ventilation and direct sunlight. At the lower end, IEEE standard 18 specifies
that capacitors shall be able to operate continuously in a −40°C ambient.
4.1
Line Compensation
As the power demand increases, the voltage drop and losses are increased since the
voltage drop is proportional to the demand current and losses are proportional to the
square of demand current. Ideal voltage profile for a transmission line is flat, while this
may not be achievable, the characteristics of the line can be modified by line
compensators so that:
i.
ii.
Ferranti effect is minimized.
Under excited operation of synchronous generators is not required.
19
iii.
The power transfer capability of the line is enhanced. Modifying the
characteristics of a line is known as line compensation.
Various compensating devices can be used as:
 Capacitors
 Capacitors and inductors
 Active voltage source (synchronous machines)
When a number of capacitors are connected in parallel to get the desired capacitance it
is known as a bank of capacitors, similarly a bank of inductors. A bank of capacitors
and/or inductors can be adjusted in steps by switching (mechanically).
Capacitors and inductors as such are passive line compensators, while synchronous
generator is an active compensator. When solid-state devices are used for switching off
capacitors and inductors, this is regarded as active compensation. Regarding the
connection type of static compensators we can have shunt and series compensation.
4.2
Serial Capacitors
Series capacitors, are capacitors connected in series with line, it is not commonly used
due to the specialized type of apparatus with a limited range of application. Also, because
of the special problems associated with each application. As shown in the figure 4.2, a
series capacitor compensates the inductive reactance. In other words, a series capacitor
is a negative reactance in series with the positive (inductive) reactance with the effect of
compensating for part or all of it. Therefore, the primary effect of the series capacitor is to
minimize, or even suppress the voltage drop caused by the inductive reactance in the
circuit. At times, a series capacitor can even be considered as a voltage regulator that
provides for a voltage boost which is proportional to the magnitude and power factor of
the through current. Therefore, a series capacitor provides a voltage rise which increases
automatically and instantly as the load grows. Also, a series capacitor produces more net
voltage rise than a shunt capacitors at a lower power factor. However, a series capacitor
betters the systems power factor much less than a shunt capacitor and has little effect on
the source current.
20
Figure 4.2 Voltage phasor diagrams for a line with a lagging power factor load (a) and (c) and leading power factor (b) and (d)
Consider the feeder circuit and its voltage phasor diagram as shown in figure 4.2
and Xc. The voltage drop through the feeder can be expressed approximately as:
∆𝑉 = 𝐼𝑅𝑐𝑜𝑠𝜑 + 𝐼𝑋𝐿 sin 𝜑
Where R is the resistance of the line, XL the inductive reactance of the line, cosφ is the
receiving end power factor and sinφ is the sin of the power factor angle.
As we can see from the phasor diagram the magnitude of the second term in the above
equation is much larger than the first. The difference gets to be much larger when the
power factor is smaller and the ratio of R/XL is small.
However when a series capacitor is applied, as shown in the figure 4.2b and 4.2c the
resultant voltage drop can be calculated as:
∆𝑉 = 𝐼𝑅𝑐𝑜𝑠𝜑 + 𝐼(𝑋𝐿 − 𝑋𝑐 ) sin 𝜑
Where XC is the capacitive reactance of the series capacitor.
The purpose of series compensation is to cancel part of the series inductive reactance of
the line using series capacitors. This helps in:
i.
increase of maximum power transfer
ii.
reduction in power angle for a given amount of power transfer
From practical point of view, it is not desirable to exceed series compensation beyond
80% of the reactance of the line. If the line it is 100% compensated than it will behave as
purely resistive element and would cause series resonance even at fundamental
21
frequency. The location of series capacitors is decided by economical factors and severity
of fault currents.
The benefits of the series capacitor compensator are associated with a problem. The
capacitive reactance XC forms a series resonant circuit with the total series reactance:
𝑋 = 𝑋𝑙 + 𝑋𝑔𝑒𝑛 + 𝑋𝑡𝑟𝑎𝑛𝑠
The natural frequency of oscillation of this circuit is given by:
𝑓𝐶 =
1
𝑋𝑐
= 𝑓√
𝑋
2 𝜋 √𝐿 𝐶
Where 𝑓 − 𝑠𝑦𝑠𝑡𝑒𝑚 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑋𝑐
− 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑐𝑜𝑚𝑝𝑒𝑛𝑠𝑎𝑡𝑖𝑜𝑛, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑟𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑒𝑑 𝑓𝑟𝑜𝑚 25% 𝑢𝑝 𝑡𝑜 75%
𝑋
For this degree of compensation 𝑓𝐶 < 𝑓 which is subharmonic compensation.
Even though series compensation has often been found to be cost-effective compared to
shunt compensation, but sustained oscillations below the fundamental system frequency
can cause the phenomenon, referred to a sub-synchronous resonance (SSR), which first
was observed in 1937, but got world-wide attention only in the 1970s, after two turbinegenerator shaft failures occurred at the Majave Generating station in Southern Nevada.
Theoretical studies pointed out that interaction between a series capacitor-compensated
line, oscillating at subharmonic frequency, and torsional mechanical oscillation of turbinegenerator set can result in negative damping with consequent mutual reinforcement of
the two oscillations. Sub synchronous resonance is often not a major problem, and low
cost counter measures and protective measures can be applied. Some of the corrective
measures are:
i.
ii.
Detecting the low levels of subharmonic currents on the line by use of sensitive
relays, which at a certain level of currents triggers the action to bypass the series
capacitors.
Modulation of generator field current to provide increased positive damping at
subharmonic frequency.
Series inductors are needed for line compensation under light load conditions to counter
the excessive voltage rise (Ferranti effect).
As the line load and, in particular the reactive power flow over the line varies, there is
need to vary the compensation for an acceptable voltage profile. The mechanical
switching arrangement for adjusting the capacitance of the capacitor bank in series with
the line is shown in figure 4.3. Capacitance is variably opening the switches of individual
capacitances with the capacitance C1, being started by a bypass switch. This is a step22
wise arrangement. The whole bank can also be bypassed by the starting switch under
any emergent conditions on the line. As the switches in series with capacitor are current
carrying suitable circuit breaking arrangement are necessary. However, breaker switched
capacitors in series are generally avoided these days the capacitors either are fixed or
thyristor switched.
Figure 4.3 Mechanical switching arrangement to adjust the capacitance of the capacitive banks
With fast advanced thyristor devices and associated switching control technology, the
series capacitance bank can be controlled much more effectively; both stepwise and
smooth control.
4.2.1 Overcompensation
Usually, the series capacitor size reactance is selected in such a way that the resultant
capacitive reactance is smaller than the reactance of the line. However, in lower voltage
levels where the resistance of the line is larger than the inductive reactance the reverse
might be preferred. The resultant condition is known as overcompensation. Figure 4.4
shows a voltage phasor diagram for overcompensation at normal load. At times, when
the selected level of overcompensation is strictly based on normal load, the resultant
overcompensation of the receiving-end voltage may not be pleasing at all because
lagging current of a large motor at start can produce a large voltage rise, as shown in
figure 4.4, which is especially harmful to lights (shortening their lives) and causes light
flicker.
23
Figure 4.4 Overcompensation of the end voltage: (a) at normal load and (b) at the start of a large motor
Figure 4.5 Voltage phasor diagram with leading power factor: (a) without series capacitors and (b) with series capacitors
As it can be seen from the figure, the voltage in the end of the line is reduced as a result
of having series capacitors.
When cos=1, sin-0 and therefore all loses are:
∆𝑈 = 𝐼𝑅
Thus, in voltage regulation, series capacitors practically have no value. That’s because
of the aforementioned reasons and others (e.g., Ferro resonance in transformers, sub
synchronous resonance during motor starting, shunting of motors during normal operation
and difficulty in the protection of capacitors from system fault current), series capacitors
do not have large application in distribution systems. However, they are employed in sub
transmission systems to modify the load division between parallel lines. For example:
 Often a new sub transmission systems to modify the load division between
parallel lines.
 Often a new sub transmission line with larger thermal capability is parallel with an
already existing line. It may be very difficult, if not impossible, to load the
subtrasmission line without overloading the old line.
24
4.3
Shunt Capacitors
The main use of shunt capacitors is in improving the power factor, which is the ratio of
the active power to the apparent power. It illustrates the relationship of the active power
that is used to produce real work, and the reactive power required to enable inductive
components such as transformers and motors to operate. As those components, with
pronounced inductive characteristics, are common in industrial facilities and power
networks, reactive power is inductive and current is lagging.
On the other hand, capacitive loads supply reactive power and the relevant current is
leading. Most loads require the two types of power to operate. The reactive current
flowing to a load will not affect the active power drawn by the load, but the supply circuit
must carry the vector sum of the active and reactive current. This later will contribute to
the power dissipated in the supply and distribution network (losses), to the voltage drop,
and to the network capacity requirements. A poor power factor will therefore result in
lower energy efficiency and reduced power quality, in some cases affecting the product
quality and the process productivity.
Many utilities employ shunt capacitor banks to help improve feeder voltage profile via
power factor correction. The capacitors, decrease demand current, decrease system
losses and reduce the voltage drop. Consequently, the voltage profile is improved. The
capacitors are connected to the feeder at specific locations at which the voltage
variation along the feeder does not violate tolerance of ± 5% of nominal voltage (figure
4.6).
Figure 4.6 Voltage profile along a feeder with capacitors
25
Shunt capacitors are, capacitors connected in parallel with lines, are used extensively in
distribution systems. Shunt capacitors supply the type of reactive power or current to
counteract the out of phase component of current required by an inductive load. In a
sense, shunt capacitors modify the characteristic of an inductive load by drawing a
leading current that counteracts some or all of the lagging component of the inductive
load current at the point of installation. Therefore, a shunt capacitor has the same effect
as an overexcited synchronous condenser, generator, or motor.
As shown in Figure 4.7, by the application of shunt capacitor to a line, the magnitude of
the source current can be reduced, the power factor can be improved, and consequently
the voltage drop between the beginning line and the load is also reduced.
However, shunt capacitors do not affect current or power factor beyond their point of
application. Figure 4.7a and c shows the single-line diagram of a line and its voltage
phasor diagram before the addition of the shunt capacitor, and Figure 4.7b and d shows
them after the addition.
Figure 4.7 Voltage phasor diagrams for a feeder circuit of a lagging power factor: (a) and (c) without compensation and (b) and
(c) with shunt capacitor compensation.
Voltage drop in lines, or in short transmission lines, with lagging power factor can be
approximated as:
∆𝑉 = 𝐼𝑅 𝑅 + 𝐼𝑋 𝑋
Where:
R is the total resistance of the feeder circuit, Ω
26
XL is the total inductive reactance of the feeder circuit, Ω
IR is the real power (or in-phase) component of the current, A
IX is the reactive component of the current lagging the voltage by 90° A
When a capacitor is installed at the receiving end of the line, as shown in figure 4.7b, the
resultant voltage drop can be calculated approximately as:
∆𝑉 = 𝐼𝑅 𝑅 + 𝐼𝑋 𝑋𝐿 − 𝐼𝐶 𝑋𝐿
Where IC is the reactive component of current leading the voltage by 90°.
Shunt compensation is more or less like load compensation. It needs to be pointed out
here that shunt capacitors/inductors cannot be distributed uniformly along the line. These
are normally connected at the end of the line and/or at midpoint of the line.
Shunt capacitors raise the load PF which greatly increases the power transmitted over
the line as it is not required to carry the reactive power. There is a limit to which transmitted
power can be increased by shunt compensation as it would require very large size
capacitor bank, which would be impractical.
When switched capacitors are employed for compensation, these should be disconnected
immediately under light load conditions to avoid large excessive voltage rise.
4.3.1
Capacitor installation types
In general, capacitors installed on feeders are pole-top banks with necessary group
fusing. The fusing applications restrict the size of the bank that can be used. Therefore,
the maximum sizes used are about 1800 kVAr at 15 kV and 3600 kVAr at higher voltage
levels. Usually, utilities do not install more than four capacitor banks (of equal sizes) on
each feeder.
Figure 4.8 illustrates the effects of a fixed capacitor on the voltage profiles of a feeder
with uniformly distributed load at heavy load and light load. If only fixed-type capacitors
are installed, as can be observed in Figure 4.8c, the utility will experience an excessive
leading power factor and voltage rise at that feeder. Therefore, some of the capacitors
are installed as switched capacitor banks so they can be switched off during light-load
conditions. Thus, the fixed capacitors are sized for light load and connected permanently.
The switched capacitors can be switched as a block or in several consecutive steps as
the reactive load becomes greater from light-load level to peak load and sized
accordingly.
27
Figure 4.8 The effects of the fixed capacitor on the voltage profile of (a) feeder with uniformity load, (b) at heavy load and (c) at
light load
This curve is called the reactive load–duration curve and is the cumulative sum of the reactive
loads (e.g., fluorescent lights, household appliances, and motors) of consumers and the reactive
power requirements of the system (e.g., transformers and regulators). Once the daily reactive
load–duration curve is obtained, then by visual inspection of the curve, the size of the fixed
28
capacitors can be determined to meet the
minimum reactive load. For example, from
figure 4.9 one can determine that the size of
the fixed capacitors required is 600 kVAar.
The remaining kVAr demands of the loads
are met by the generator or preferably by the
switched capacitors. However, since
meeting the kVAr demands of the system
from the generator is too expensive and may
create problems in the system stability,
capacitors are used. Capacitor sizes are
selected to match the remaining load
characteristics from hour to hour.
Many utilities apply the following rule of
thumb to determine the size of the switched
capacitors:
Add switched capacitors until:
𝑘𝑉𝐴𝑟 𝑓𝑟𝑜𝑚 𝑠𝑤𝑖𝑡𝑐ℎ𝑒𝑑 + 𝑓𝑖𝑥𝑒𝑑 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟𝑠
𝑘𝑉𝐴𝑟 𝑜𝑓 𝑝𝑒𝑎𝑘 𝑟𝑒𝑎𝑐𝑡𝑖𝑣𝑒 𝑓𝑒𝑒𝑑𝑒𝑟 𝑙𝑜𝑎𝑑
≥ 0.70
From the voltage regulation point of view, the
kilovars needed to raise the voltage at the
end of the line to the maximum allowable
voltage level at minimum load (25% of peak
load) are the size of the fixed capacitors that
should be used. On the other hand, if more
than one capacitor bank is installed, the size
of each capacitor bank at each location
should have the same proportion. However, the resultant voltage rise must not exceed the lightload voltage drop. The approximate value of the percent voltage rise can be calculated from:
∆𝑈% =
𝑄𝐶,3𝑓 𝑥𝑙 𝑙
10 𝑉𝑙2
Where:
U%
is the percent voltage rise
Qc,3f
is the three-phase reactive power due to fixed capacitors applied, kVAr
xl
is the line reactance, Ω/min
29
l
is the length of feeder from sending end of feeder to fixed capacitor location, min
Vl
is the line-to-line voltage, kV
Some utilities use the following rule of thumb: The total amount of fixed and switched capacitors
for a feeder is the amount necessary to raise the receiving-end feeder voltage to maximum at
50% of the peak feeder load. Once the kilovars of capacitors necessary for the system are
determined, there remains only the question of proper location. The rule of thumb for locating
the fixed capacitors on feeders with uniformly distributed loads is to locate them approximately
at two-thirds of the distance from the substation to the end of the feeder.
For the uniformly decreasing loads, fixed capacitors are located approximately halfway out on
the feeder. On the other hand, the location of the switched capacitors is basically determined by
the voltage regulation requirements, and it usually turns out to be the last one-third of the feeder
away from the source.
4.4 Power factor correction
Figure 4.9 Sizing of the fixed and switched capacitors to meet the
A typical utility system would have a reactive daily reactive power demand
load at 0.80 power factor during the summer
months. Therefore, in typical distribution loads, the current lags the voltage, as shown in
figure 4.10a. The cosine of the angle between current and sending voltage is known as
the power factor of the circuit. If the in-phase and out-of-phase components of the current
I are multiplied by the receiving-end voltage on the line V2, the resultant relationship can
be shown on a triangle known as the power triangle, as in figure 4.10b. This figure shows
the triangular relationship that exists between kilowatts, kilovoltamperes, and kilovars.
Figure 4.10 (a) Phasor diagram and (b) Power triable for a typical distribution load
Note that, by adding the capacitors, the reactive power component Q of the apparent
power S of the load can be reduced or totally suppressed. Figures 4.11a and 4.12
illustrate how the reactive power component Q increases with each 10% change of power
factor. Figure 4.11a also illustrates how a portion of lagging reactive power Q old is
cancelled by the leading reactive power of capacitor Qc.
Note that, as illustrated in figure 4.11, even a 0.80 power factor of the reactive power
(kVAr) size is quite large, causing a 25% increase in the total apparent power (kVA) of
30
the line. At this power factor, 75 kVAr of capacitors is needed to cancel out the 75 kVAr
of the lagging component.
As previously mentioned, the generation of reactive power at a power plant and its supply
to a load located at a far distance is not economically feasible, but it can easily be provided
by capacitors (or overexcited synchronous motors) located at the load centers. Figure
4.12 illustrates the power factor correction for a given system.
Figure 4.11 Illustration of (a) the use of a power triangle for the power factor correction by employing capacitive reactive power
and (b) the required increase in the apparent and reactive powers as a function of the load power factor, holding the real power
of the load constant.
As illustrated in the figure, capacitors draw leading reactive power from the source; that
is, they supply lagging reactive power to the load. When a shunt capacitor of QC kVA is
installed at the load, the power factor can be improved from:
31
cosφ1 =
𝑃
√𝑃2 + 𝑄12
Up to:
cosφ1 =
𝑃
√𝑃2 + 𝑄22
=
𝑃
√𝑃2 + (𝑄1 − 𝑄𝐶 )2
Where
Figure 4.12 Illustration of power factor correction by employing a shunt capacitor
5 Questions
1. Who are the main producers and consumers of reactive power?
Chapter 2
2. What are the main factors of voltage drop?
Chapter 3.1
3. What are the advantages and disadvantages of series capacitor banks?
Chapter 4.2
4. What are the advantages and disadvantages of shunt capacitor banks?
Chapter 4.3
5.
6 Homework task
For the industrial consumer that is connected directly to the power line of 110 kV, it is
needed to determine the power of the capacitive bank connected in parallel with near the
consumer in order to correct the power factor from 0.8 to 0.95. Determine also the
increase of voltage after the correction.
The impedance of the line is zu=0.2+j4 [Ω/km], the admittance is yu=0+j 2.7x10-6 [Ω/km],
length of the line is l= 50 [km], the voltage at the beginning U 1=111.6 [kV], at the end of
the line U2=104 [kV] with a connected load S=40 [MVA] and cos=0.8 (ind).
32
Solution:
The apparent power is 𝑆1 = 𝑆1 (𝑐𝑜𝑠1 + 𝑗𝑠𝑖𝑛1 ) = 40(0.8 + 𝑗0.6) = (30 + 𝑗24) [𝑀𝑉𝐴]
According to the power triangle in the figure 5.1 we can write:
𝑄1 = 𝑃1 𝑡𝑔1
We can see that in order to achieve a better power factor we have to decrease the angle
 from 1 to 2, by knowing the angle 2 we can easily determine the reactive power flow
in the line after compensation as:
𝑄2 = 𝑃1 𝑡𝑔2 or cos 𝜃2 =
𝑃1
𝑃
√𝑃12 +(𝑄2 −𝑄𝐶 )2
= 𝑆1 = 0.95
2
Therefore the power of the compensator to correct the power factor is:
𝑄𝑐 = 𝑄1 − 𝑄2 = 𝑃1 (𝑡𝑔𝜃1 − 𝑡𝑔𝜃2 ) = 32(0.75 − 0.328) = 13.48 [𝑀𝑉𝐴𝑟]
𝑄𝑐 = −𝑗13.48 [𝑀𝑉𝐴𝑟]
We can conclude that the after the compensation the power that is withdrawn from the
load is:
𝑆2 = 𝑃1 + 𝑗𝑄2 = (32 + 𝑗24) + (−𝑗13.48)
= (32 + 𝑗10.52) [𝑀𝑉𝐴]
To determine the voltage at the end of the line after
compensation we suppose that the voltage in the
beginning is constant. The power line we represent
with the  model as below:
33
Where the parameters of the line are:
Figure 6.1 Power triangle
𝑍𝐿 = (0.2 + 𝑗0.4) ∗ 50 = (10 + 𝑗20) [𝛺]
𝑌𝐿 = (0 + 𝑗 27 10−6 ) ∗ 50 = 𝑗135 10−6 [𝑆]
Now we calculate the voltage at the end point after compensation according to the
equivalent scheme of the power line, and since the active power does not change we can
write:
𝑈1 = 𝑈2𝑎𝑓𝑡𝑒𝑟 +
𝑃1 𝑅𝐿 + 𝑄2 ′𝑋𝐿
𝑃1 𝑋𝐿 − 𝑄2 ′𝑅𝐿
+𝑗
𝑈2𝑎𝑓𝑡𝑒𝑟
𝑈2𝑎𝑓𝑡𝑒𝑟
Because of the Ferranti effect the reactive power at the end of the line will change as
some part of the reactive power will be compensated by the lines with a value of QC2:
𝑄2′ = 𝑄1 − 𝑄𝑐2 = 10.52 − 𝑈2𝑎𝑓𝑡𝑒𝑟 67.5 ∗ 10−6
2
𝑈12
𝑃1 𝑅𝐿 + (10.52 − 𝑈2𝑎𝑓𝑡𝑒𝑟 67.5 ∗ 10−6 )𝑋𝐿
= [𝑈2𝑎𝑓𝑡𝑒𝑟 +
]
𝑈2𝑎𝑓𝑡𝑒𝑟
2
𝑃1 𝑋𝐿 − (10.52 − 𝑈2𝑎𝑓𝑡𝑒𝑟 67.5 ∗ 10−6 )𝑅𝐿
+[
]
𝑈2𝑎𝑓𝑡𝑒𝑟
By having only the U2after as unknown we will obtain:
4
2
𝑈2𝑎𝑓𝑡𝑒𝑟
− 11441 𝑈2𝑎𝑓𝑡𝑒𝑟
+ 56961.4 = 0
After the calculation the real solution is:
𝑈2𝑎𝑓𝑡𝑒𝑟 = 106.73
In the case of the compensation of the power factor up to 0.95 the voltage will increase
for:
∆𝑈2 = 𝑈2𝑎𝑓𝑡𝑒𝑟 − 𝑈2𝑏𝑒𝑓𝑜𝑟𝑒 = 106.73 − 104 = 2.73 𝑘𝑉
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7 Conclusion
Reactive power control compensation devices are designed to ensure stable levels of
electric power voltage by maintaining predetermined voltage levels at network control
points. In some cases, especially for long-distance power transmission, these devices are
also expected to maintain predetermined levels of static and dynamic stability of electric
power systems and load stability.
The reactive power transfer through the lines can be reduced with different compensation
methods. By the compensation the reactive power we will increase the total power
capability transfer and reduce the total current which means reducing also the voltage
drops.
In practice these static compensating devices are mostly shunt capacitor banks and
shunting reactors that perform graduated control of reactive power, reactor groups
commutated by modern switches, which can be discussed as an extension of this seminar
paper.
8 References
[1]
[4]
[5]
T. Gonen. Electric Power Distribution Engineering-Third Edition. CRC Press
2014
T. A. Short. Electric Power Distribution Equipment and Systems. CRC Press,
2014
G. Pula. Elektroenergjetika. Fakulteti i Inxhinierise Elektrike dhe Kopjuterike
Prishtine. 1986
G. Pula. Bartja dhe Shperndarja e Energjise Elektrike, Prishtine 1980
D. P. Kothari. Modern Power Systems. Third Edition 2003
[6]
[7]
J. Kock and C. Strauss. Practical Power Distribution for Industry, 2004
A. A. Sallam. Electric Distribution Systems. Wiley 2011
[2]
[3]
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